by Woolhiser and Liggett
34
and by Lin et al.
24
. Chow and Ben-Zvi
12
and Kawahara and Yokoyama
22
developed two- dimensional overland flow models for impervious surfaces
based on the Lax–Wendroff method
23
and a finite-element method, respectively. Zhang and Cundy
37
developed a two- dimensional overland flow model for a temporally constant,
but spatially varying infiltration. MacCormack finite-differ- ence method was used to obtain the numerical solution.
Tayfur et al.
31
also developed a two-dimensional overland flow model based on an implicit method
8
for solving the governing equations and applied the model to experimental
hill slopes. It was reported that consideration of microtopo- graphy in fine detail could result in numerical instability and
convergence problems.
All the overland flow models discussed so far describe the infiltration process in an empirical way. Smith and
Woolhiser
29
were probably the first to develop a con- junctive model for overland flow. They solved the one-
dimensional Richards equation for unsaturated flow in the subsurface together with the one-dimensional kinematic
wave equation for the surface flow. The Richards equation was solved by an implicit Crank–Nicolson finite-difference
formulation. Akan and Yen
5
developed a sophisticated con- junctive model in which they solved the complete Saint-
Venant equations and the two-dimensional Richards equation. The Saint-Venant equations were solved by the
four-point implicit method, while Richards equation was solved by the SLOR technique. In the very versatile, com-
plete catchment model, SHE
2
, the overland flow component is simulated using the diffusion wave approximation. The
subsurface flow in the unsaturated zone is represented by the one-dimensional Richards equation. The surface flow equa-
tions are solved by the explicit procedure developed by Preissmann and Zaoui
28
, while the Richards equation is solved by the implicit finite-difference scheme. Govindaraju
and Kavvas
16
, in a detailed study of hillslope hydrology through a stream flow—overland flow—subsurface flow
model, used the diffusion wave approximation for the sur- face flow component. The surface flow equation was solved
by the centred implicit scheme, while the two-dimensional Richards equation was solved by the SLOR technique.
In this paper, an alternative conjunctive numerical model is presented for simulating the overland flow. The numerical
model is based on a conjunctive solution of the complete Saint-Venant equations for the surface flow and the two-
dimensional Richards equation. Presently available full dynamic models for the overland flow employ classical
second-order accurate finite-difference methods for solving the Saint-Venant equations. These methods often result in
high frequency oscillations when there are sharp changes in the flow parameters and require ad hoc procedures for
obtaining smooth solutions
6
. In this study, a simple to implement
high-resolution essentially
non-oscillating ENO scheme
27
is applied to solve the surface flow com- ponent in the conjunctive model. The explicit ENO scheme
also results in a simple method for connecting the surface and subsurface flow components. Presently available
conjunctive models for the overland flow use Richards equation in pressure-head form, which may result in signi-
ficant mass balance error for complex subsoil conditions. Many studies
7,19,9
have shown that the ‘mixed form’ of the Richards equation results in better numerical behaviour
than the other forms. Therefore, in this study, the two- dimensional Richards equation in the mixed form is used
for simulating the subsurface component. This equation is solved by a recently developed strongly implicit procedure
20
. This is computationally more efficient than the SLOR technique adopted in the currently available conjunctive
models. The proposed numerical model is verified using the experimental data available in the literature
29
. Robust- ness of the numerical model is demonstrated by considering
a test case in which a clay lens is present in the subsurface. One of the objectives of developing a sophisticated model is
that it can act as a standard of comparison for simplified models. Therefore, the proposed numerical model is used to
study the errors that may result in the prediction of the out- flow hydrograph if a one-dimensional simulation of the sub-
surface flow is used.
2 GOVERNING EQUATIONS
Mathematical modeling of overland flow involves the solu- tion of the governing equations for both the surface flow and
the groundwater flow with seepage at the ground surface acting as the connecting link. In the present study Fig. 1,
the surface runoff is represented by the one-dimensional flow equations in the x-direction, while the groundwater
flow is represented by the two-dimensional Richards equa- tion in the x and z directions.
2.1 Surface flow equations
Surface flow is assumed to occur in a prismatic channel of rectangular section. The one-dimensional shallow water
Fig. 1. Definition sketch for a conjunctive model.
568 V. Singh, S. M. Bhallamudi
flow equations in conservation form for such a case are given by:
] U
] t
þ ]
F ]
x ¼
S 1
in which, U ¼
h q
, F ¼
q q
2
h þ
gh
2
2 8
: 9
= ;
, S ¼
R ¹ I gh
S ¹
S
f
2 In eqns 1 and 2: h ¼ flow depth m; q ¼ discharge per
unit width m
2
s; g ¼ acceleration due to gravity ms
2
; R ¼
volumetric rate of rainfall per unit surface area ms; I ¼ volumetric rate of infiltration per unit area ms; S
¼ bottom slope in the direction of flow; S
f
¼ friction slope;
x ¼ distance along the flow direction m; and t ¼ time s. The assumptions and the derivation of the above equations
have been reported earlier
36,1,11
and are not repeated here. The friction slope, S
f
is computed using the Darcy– Weisbach formula taking into account the effect of rainfall
on frictional resistance. S
f
¼ f
d
q
2
8gh
3
3 in which, f
d
¼ frictional resistance coefficient. Evaluation
of f
d
depends on the instantaneous state of flow. The flow is laminar in all the studies reported here and, therefore, f
d
is given by the following equation
5
. f
d
¼ C
L
Re 4
where Re ¼ Reynolds number ¼ qv v ¼ kinematic viscosity of water, and C
L
is a constant which depends on the rainfall intensity.
2.2 Subsurface flow equations
In the present study, subsurface flow is modeled as two- dimensional motion of a single-phase incompressible fluid
in an incompressible porous medium. The two-dimensional, transient flow equation in an isotropic porous medium is
derived by applying the principle of conservation of mass. This equation without sources and sinks within the flow
domain can be written as
] v
] t
þ ]
V
x
] x
þ ]
V
z
] z
¼ 5
where, v ¼ volumetric moisture content; V
x
and V
z
¼ darcy
flow velocity in the x and z direction, respectively, and x and z are distances along the two coordinate directions;
z is taken positive downwards. It is assumed that the Darcy’s law is applicable for evaluating the velocity
components.
V
x
¼ ¹ K
w ]
w ]
x ,
V
z
¼ ¹ K
w ]
w ]
z ¹
1:0 6
in which, w ¼ pressure head m and Kw ¼ unsaturated hydraulic conductivity ms which depends on the pressure
head, w. Substitution of eqn 6 in eqn 5 yields the Richards equation
14
: ]
v ]
t ¼
] ]
x K
w ]
w ]
x þ
] ]
z K
w ]
w ]
z ¹
1:0 7
eqn 7 is said to be in ‘mixed form’ since it includes both the dependent variables v and w. Hydraulic relationships
between the pressure head, w, and the hydraulic conductiv- ity, K w–K relationship, and between the moisture con-
tent, v, and the pressure head w w–v relationship, are needed while solving eqns 5 and 6 in the unsaturated
zone. In general, w–K and w–v relationships are not unique and soils exhibit different behaviour during
wetting and drying phases. This hysteresis in soil charac- teristics is not considered for the cases studied in the
present work. However, the hysteresis can be included by employing different w–K and w–v relationships for wet-
ting and drying processes. Although semi-analytical equations are available for describing w–K and w–v rela-
tionships, the soil characteristics derived from the experi- mental and field data are employed wherever available.
These relationships are specific to the case studied and are described later.
3 NUMERICAL SOLUTION 3.1 Surface flow
Surface flow equations constitute a set of nonlinear hyper- bolic partial differential equations. A recently developed
high-resolution essentially non-oscillating ENO scheme for solving the shallow water flow equations
27
is employed in the present study to solve the surface flow equations. The
scheme, proposed by Nujic
27
, is modified suitably to account for the non-zero source term in the continuity equa-
tion. This scheme, unlike many other classical second-order accurate schemes such as the MacCormack method, is non-
oscillatory even when sharp gradients in the flow variables are present. The main advantages of this scheme are its
simplicity, ease of implementation and ease of extension to a two-dimensional case. It is also very attractive for the
present application because it is possible to account for the variable bottom topography in a convenient and accurate
way. It is an explicit, two-step predictor–corrector scheme which results in second-order accuracy in both space and
time. Only a brief description of the method as applied in the present study is outlined here.
3.1.1 Predictor part The finite-difference analog of eqn 1 is written here for a
finite-difference grid where the subscript i refers to the grid point in the x direction and the superscripts n and refer to
the values at the known time level and the predictor level, respectively. The finite-difference form of eqn 1 for the
Conjunctive surface–subsurface modeling of overland flow 569
explicit determination of U
i
is written as U
p i
¼ U
n i
¹ Dt
Dx F
n i þ 1=2
¹ F
n i ¹ 1=2
þ DtS
n i
8 where F
iþ12
n represents the numerical flux through the cell face between nodes i þ 1 and i. Dx is the grid spacing and
Dt is the computational time step. All the terms on the right-hand side of eqn 8 are evaluated at the known
time level n and, therefore, U
i
can be computed explicitly. The numerical flux F
iþ12
is computed using the following formula.
F
i þ 1
2
¼ 1
2 [
F
R
þ F
L
¹ a
U
R
¹ U
L
] 9
in which, a ¼ a positive coefficient, F
R
¼ fU
R
the flux computed using the information from the right side of the
cell face and F
L
¼ fU
L
¼ the flux computed using the information from the left side of the cell face. U
L
and U
R
are obtained using a MUSCL monotone upwind scheme for conservation laws approach.
U
L
¼ U
i
þ dU
i
2 10
U
R
¼ U
i þ 1
¹ dU
i þ 1
2 11
There are several ways of determining dU
i
and dU
iþ1
using different
slope limiter
procedures
6,35
. The
‘minmod’ limiter is followed in the present study. Accord- ing to this,
dU
i
¼ minmod
U
i þ 1
¹ U
i
, U
i
¹ U
i ¹ 1
12 where the minmod function is defined as
minmod a
, b
¼ a
if lal , lbl and ab . 0
b if
lbl , lal and ab . 0 if ab 0
8 :
13 The positive coefficient a is determined using the maxi-
mum value for all the grid points of the largest eigen value of the Jacobian of the system of equations. This is
given as
a max q
i
h
i
þ
gh
i
p i ¼ 1 to N
14 where, N is the total number of grid points. The vector
equation eqn 8 gives the predicted values of h and q at the unknown time level at any node i.
3.1.2 Corrector part The vector U at the unknown time level n þ 1 and at node i,
i.e. U
i nþ1
is computed using the predicted values and the values at the time level n.
U
n þ 1 i
¼ 0:5 U
n i
þ U
p i
¹ Dt
Dx F
p i þ 1=2
¹ F
p i ¹ 1=2
ÿ þ DtS
p i
15 where
F
p i þ 1=2
¼ 1
2 [
F
p R
þ F
p L
¹ a
U
p R
¹ U
p L
] 16
Following the recommendations of Alcrudo et al.
6
, U
R
and U
L
are determined from U
iþ1
and U
i
using the same dU
i þ 1
and dU
i
as determined in the predictor step. This procedure results in better numerical stability.
In eqn 15, only the source term in the momentum equa- tion the second component of the vector equation is eval-
uated using the predicted values of h and q. However, the source term in the continuity equation i.e. R ¹ I
i
is eval- uated using the values at the known time level instead of the
predicted values. Strictly speaking, this procedure decou- ples the subsurface flow computations and the surface
flow computations during the computational time step Dt. However, the response of the subsurface flow to the varia-
tion in surface flow depth is much slower than the response of surface flow to changes in the rate of infiltration
5
. There- fore, the above decoupling does not affect the results sig-
nificantly. In fact, numerical experimentation showed that determination of the infiltration rate, I, during the corrector
step by using the predicted flow depth did not alter the results. On the other hand, the decoupling procedure
resulted in significant savings of the computational time, since the subsurface flow is computed only once during a
time step.
3.2 Initial and boundary conditions
